Central Limit Theorems for Estimators in ARMA Models under Weak Dependence: Theory and Empirical Validation

Abstract We establish Central Limit Theorems (CLTs) for Quasi–Maximum Likelihood Estimators (QMLE) in ARMA(p,q) models under weak dependence, formalized via α-mixing. The estimators are √n–consistent and asymptotically normal, with covariance in the familiar HAC–sandwich form. To assess finite–sample behavior, we run Monte Carlo experiments for ARMA(1,1) under Gaussian and non-Gaussian innovations (Student-t with ν=3,5, Gamma with α=2, and Poisson with λ=2). Histograms and Q--Q plots of standardized statistics align closely with N(0,1), empirical coverages approach 95%, and joint scatters of (√n(φ-φ₀), √n(θ-θ₀)) exhibit the expected elliptical shape. The results confirm that QMLE with HAC inference provides reliable estimation and valid uncertainty quantification beyond the classical i.i.d.\ Gaussian setting, offering a practical framework for weakly dependent, non-Gaussian time series.